Finite Group Databasebeta

← #13105985

Subgroups — #13105985

Order 1600.

704 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#87900C211yesmaximal, characteristic
2C2×#87872C211yesmaximal, characteristic, complemented
3C5×#4070C212yesmaximal, complemented
4#11717093C212yesmaximal, complemented
5#11717071C212yesmaximal, complemented
6#11717016C212yesmaximal, complemented
7#11716990C212yesmaximal, complemented
8#11716967C211yesmaximal, characteristic, complemented
9#11716958C212yesmaximal, complemented
10C22×C4×C52C2211yescharacteristic
11C2×C22⋊(C5⋊D5)C2211yescharacteristic
12C2×C22⋊(C5⋊D5)C2211yescharacteristic
13#87900—24no—
14C2×C4⋊(C5⋊D5)C2211yescharacteristic
15C2×C4×C5⋊D5C2211yescharacteristic
16C2×C4×C5⋊D5C2211yescharacteristic
17C2×#6045C2211yescharacteristic
18C2×C5×Dic10C2212yes—
19C2×#6039C2212yes—
20C2×#6036C2212yes—
21#87872—24no—
22C5×C22⋊Dic5C2214yes—
23C5×C4⋊Dic5C2212yes—
24C5×#683C2214yes—
25#87780C2214yes—
26#87779C2212yes—
27#87779C2214yes—
28Dic52C2212yes—
29#65962—52nomaximal
30C23×C52C2311yescharacteristic
31C22×C5⋊D5C2311yescharacteristic
32C22×C5⋊D5C2311yescharacteristic
33C2×C4×C52—22no—
34C2×C4×C52C2311yescharacteristic
35C2×C4×C52C2311yescharacteristic
36C22⋊(C5⋊D5)—22no—
37C22⋊(C5⋊D5)—22no—
38C2×#1039C2311yescharacteristic
39C2×#1039C2311yescharacteristic
40C4⋊(C5⋊D5)—22no—
41C4×C5⋊D5—22no—
42C4×C5⋊D5D414yes—
43#6045—22no—
44C2×C5×Dic5C2314yes—
45C2×C5×Dic5C2314yes—
46C5×Dic10—24no—
47#6039—24no—
48#6036—24no—
49C2×Q8×D5—52no—
50C2×#709—52no—
51C2×#709—102no—
52C5×C22⋊Q8—52no—
53#4092—52no—
54#4070D512yescomplemented
55#4043—52no—
56#4040—54no—
57#4038—52no—
58#4034—54no—
59#4031—54no—
60#4029—54no—
61C22×C52—21no—
62C22×C52—22no—
63C22×C52C2411yescharacteristic
64C2×C5⋊D5—22no—
65C2×C5⋊D5—21no—
66C2×C5⋊D5—22no—
67C2×C5⋊D5C2×D412yes—
68C4×C52—21no—
69C4×C52C2×D412yes—
70#1039—21no—
71#1039C2×D412yes—
72C5×Dic5—24no—
73C5×Dic5—24no—
74C2×C5×Q8—52no—
75C22×C4×C5—22no—
76C22×C4×C5C2×D512yes—
77C2×C22⋊D5—52no—
78C2×C22⋊D5—52no—
79C2×C22⋊D5—102no—
80C2×C22⋊D5—102no—
81Q8×D5—108no—
82#709—108no—
83#709—1016no—
84C2×D20—52no—
85C2×D20—102no—
86C2×C4×D5—52no—
87C2×C4×D5—52no—
88C2×C4×D5—54no—
89C2×C4×D5—102no—
90C2×C4×D5—102no—
91C2×Dic10—52no—
92C2×Dic10—54no—
93C2×Dic10—102no—
94C2×Dic10C2×D512yes—
95C5×C4⋊C4—54no—
96C5×C4⋊C4—52no—
97C5×C22⋊C4—54no—
98C22⋊Dic5C2×D514yes—
99#685—54no—
100#685—54no—
101#685—54no—
102C4⋊Dic5—54no—
103C4⋊Dic5C2×D512yes—
104#683—54no—
105#683C2×D514yes—
106C4×Dic5—54no—
107#536—251noSylow
108C2×C52—41no—
109C2×C52C22×D411yescharacteristic
110C2×C522−1+412yes—
111C5⋊D5—22no—
112C5⋊D5—41no—
113C23×C5—22no—
114C23×C5C22×D512yes—
115C22×D5—52no—
116C22×D5—52no—
117C22×D5—102no—
118C22×D5—102no—
119C5×Q8—104no—
120C2×C4×C5—28no—
121C2×C4×C5—24no—
122C2×C4×C5—22no—
123C2×C4×C5—22no—
124C2×C4×C5—54no—
125C2×C4×C5—54no—
126C2×C4×C5C22×D512yes—
127C2×C4×C5C22×D512yes—
128C22⋊D5—104no—
129C22⋊D5—104no—
130C22⋊D5—108no—
131C22⋊D5—108no—
132C2×Dic5—52no—
133C2×Dic5—52no—
134C2×Dic5—102no—
135C2×Dic5—102no—
136C2×Dic5C22×D514yes—
137C2×Dic5C22×D514yes—
138D20—104no—
139D20—108no—
140C4×D5—58no—
141C4×D5—108no—
142C4×D5—108no—
143C4×D5—108no—
144C4×D5—104no—
145Dic10—24no—
146Dic10—108no—
147Dic10—104no—
148Dic10—108no—
149C2×#41—251no—
150C22×Q8—251no—
151C4⋊Q8—252no—
152#124—252no—
153#123—252no—
154#123—252no—
155C22⋊Q8—52no—
156C22⋊Q8—252no—
157#117—251no—
158C52#53611yescharacteristic, complemented, Sylow
159C22×C5—24no—
160C22×C5—22no—
161C22×C5—22no—
162C22×C5—22no—
163C22×C5—22no—
164C22×C5—44no—
165C22×C5C23×D512yes—
166C2×D5—54no—
167C2×D5—104no—
168C2×D5—102no—
169C2×D5—102no—
170C2×D5—104no—
171C2×D5—102no—
172C2×D5—102no—
173C2×D5—102no—
174C2×D5—204no—
175C2×D5—204no—
176C4×C5—24no—
177C4×C5—22no—
178C4×C5—24no—
179C4×C5—104no—
180C4×C5—104no—
181C4×C5D4×D514yes—
182Dic5—24no—
183Dic5—24no—
184Dic5—54no—
185Dic5—104no—
186Dic5—104no—
187Dic5—102no—
188#41—504no—
189C2×Q8—52no—
190C2×Q8—252no—
191C2×Q8—251no—
192C2×Q8—504no—
193C2×D4—251no—
194C2×D4—251no—
195C2×D4—251no—
196C22×C4—252no—
197C22×C4—251no—
198C22×C4—251no—
199C22×C4D5211yescharacteristic
200C4⋊C4—52no—
201C4⋊C4—54no—
202C4⋊C4—254no—
203C22⋊C4—54no—
204C22⋊C4—252no—
205C22⋊C4—254no—
206C42—252no—
207C2×C5—24no—
208C2×C5—22no—
209C2×C5—44no—
210C2×C5—42no—
211C2×C5C2×D4×D512yes—
212C2×C5#414712yes—
213C2×C5#415012yes—
214D5—104no—
215D5—204no—
216D5—204no—
217D5—202no—
218C23—251no—
219C23—251no—
220C23C2×D5211yescharacteristic
221Q8—104no—
222Q8—502no—
223Q8—504no—
224D4—502no—
225D4—502no—
226D4—502no—
227C2×C4—22no—
228C2×C4—54no—
229C2×C4—54no—
230C2×C4—251no—
231C2×C4—254no—
232C2×C4—251no—
233C2×C4—504no—
234C2×C4—502no—
235C2×C4C2×D5211yescharacteristic
236C2×C4C2×D5211yescharacteristic
237C5—22no—
238C5#6596212yescomplemented
239C22—22no—
240C22—21no—
241C22—252no—
242C22—502no—
243C22—502no—
244C22—501no—
245C22C22×D5211yescharacteristic
246C4—21no—
247C4—104no—
248C4—104no—
249C4—252no—
250C4—501no—
251C4#8788612yes—
252C2—41no—
253C2—502no—
254C2—1001no—
255C2#1171746212yes—
256C2C2×#8788611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D52
commutator, socle, cosocleC22×C52C24
FittingC22×C4×C52C22
prime radicalC2×#87900C2
hypercenterC22×C4D52
hypocenterC52#536
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C22×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#87872C2semidirect
3. C5×#4070C2semidirect
4. #11717093C2semidirect
5. #11717071C2semidirect
6. #11717016C2semidirect
7. #11716990C2semidirect
8. #11716967C2semidirect
9. #11716958C2semidirect
54. #4070D5semidirect
158. C52#536semidirect
238. C5#65962semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#879002
C2×#878722
C5×#40702
#117170932
#117170712
#117170162
#117169902
#117169672
#117169582
#659625